Question

name teacher hour

1. use synthetic division to find the quotient and remainder. write the result as

dividend = (divisor)(quotient) + remainder.
$(x^3 + 5x^2 + 5x - 2) \div (x + 1)$

1. use long division to find the quotient and remainder. write the result as

dividend = (divisor)(quotient) + remainder.
$(x^3 - 4x^2 - 4x + 16) \div (x^2 - 4)$

1. use synthetic substitution to find $p(-3)$ for $p(x) = x^4 + 4x^3 - 16x - 16$.

1. determine whether the given binomial is a factor of $p(x)$. explain. then write the remaining

factor.
$(x + 1); (x^2 + 2x + 1)$

explanation: circle one: yes or no
remaining factor:
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algebra 2b unit 5

Explanation:

Question 8 (Synthetic Division)

Step 1: Identify coefficients and root

The dividend is \(x^3 + 5x^2 + 5x - 2\), so coefficients are \(1, 5, 5, -2\). The divisor is \(x + 1\), so root \(r=-1\).

Step 2: Set up synthetic division

\[

$$\begin{array}{r|rrrr} -1 & 1 & 5 & 5 & -2 \\ & & -1 & -4 & -1 \\ \hline & 1 & 4 & 1 & -3 \\ \end{array}$$

\]

Step 3: Determine quotient and remainder

Quotient: \(x^2 + 4x + 1\), Remainder: \(-3\)

Step 4: Write in dividend = (divisor)(quotient) + remainder

\(x^3 + 5x^2 + 5x - 2=(x + 1)(x^2 + 4x + 1)-3\)

Step 1: Divide leading terms

Divide \(x^3\) by \(x^2\) to get \(x\). Multiply divisor \(x^2 - 4\) by \(x\): \(x(x^2 - 4)=x^3 - 4x\). Subtract from dividend: \((x^3 - 4x^2 - 4x + 16)-(x^3 - 4x)=-4x^2 + 0x + 16\).

Step 2: Divide next leading term

Divide \(-4x^2\) by \(x^2\) to get \(-4\). Multiply divisor by \(-4\): \(-4(x^2 - 4)=-4x^2 + 16\). Subtract: \((-4x^2 + 0x + 16)-(-4x^2 + 16)=0\).

Step 3: Determine quotient and remainder

Quotient: \(x - 4\), Remainder: \(0\)

Step 4: Write in dividend = (divisor)(quotient) + remainder

\(x^3 - 4x^2 - 4x + 16=(x^2 - 4)(x - 4)+0\)

Step 1: Identify coefficients and root

\(p(x)=x^4 + 4x^3 + 0x^2 - 16x - 16\), coefficients: \(1, 4, 0, -16, -16\). Root \(r=-3\) (for \(p(-3)\)).

Step 2: Set up synthetic substitution

\[

$$\begin{array}{r|rrrrr} -3 & 1 & 4 & 0 & -16 & -16 \\ & & -3 & -3 & 9 & 21 \\ \hline & 1 & 1 & -3 & -7 & 5 \\ \end{array}$$

\]

Step 3: Result

The last number is the value of \(p(-3)\), so \(p(-3)=5\).

Answer:

Quotient: \(x^2 + 4x + 1\), Remainder: \(-3\); \(x^3 + 5x^2 + 5x - 2=(x + 1)(x^2 + 4x + 1)-3\)

Question 9 (Long Division)